Theorem sbcgf 2881

Description: Substitution for a variable not free in a wff does not affect it. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypothesis

Ref	Expression
sbcgf.1	|- F/ x ph

Assertion

Ref	Expression
sbcgf	|- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Proof of Theorem sbcgf

Step	Hyp	Ref	Expression
1		sbcgf.1	. 2 |- F/ x ph
2		sbctt 2880	. 2 |- ( ( A e. V /\ F/ x ph ) -> ( [. A / x ]. ph <-> ph ) )
3	1, 2	mpan2 415	1 |- ( A e. V -> ( [. A / x ]. ph <-> ph ) )

Syntax hints:   -> wi 4   <-> wb 103   F/wnf 1389   e. wcel 1433   [.wsbc 2815

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816

This theorem is referenced by:  sbc19.21g  2882  sbcg  2883  sbcabel  2895